On singular pencils with commuting coefficients
Vadym Koval, Patryk Pagacz · Linear and Multilinear Algebra · 2024
We investigate the relation between the spectrum of matrix (or operator) polynomials and the Taylor spectrum of its coefficients. We prove that the matrix polynomial with commuting coefficients is singular, i.e. its spectrum is the whole complex plane, if and only if (0,0,…,0) belongs to the Taylor spectrum of its coefficients. On the other hand, we prove that this equivalence is no longer true if we consider the operators on infinite dimensional Hilbert space as coefficients of polynomial. As a consequence, we could propose a new description of the (Taylor) spectrum of k-tuple of matrices and we could disprove the conjecture previously proposed in the literature. Additionally, we pointed out the Kronecker forms of the pencils with commuting coefficients.